TY - JOUR A1 - Clavier, Pierre J. A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - An algebraic formulation of the locality principle in renormalisation JF - European Journal of Mathematics N2 - We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler-Maclaurin formula on lattice cones, we renormalise the exponential generating function which sums over the lattice points in a lattice cone. As a consequence, for a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate germs. KW - Locality KW - Renormalisation KW - Algebraic Birkhoff factorisation KW - Partial algebra KW - Hopf algebra KW - Rota-Baxter algebra KW - Multivariate meromorphic functions KW - Lattice cones Y1 - 2019 U6 - https://doi.org/10.1007/s40879-018-0255-8 SN - 2199-675X SN - 2199-6768 VL - 5 IS - 2 SP - 356 EP - 394 PB - Springer CY - Cham ER - TY - CHAP A1 - Clavier, Pierre J. A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - Renormalisation and locality BT - branched zeta values T2 - Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2 Y1 - 2020 SN - 978-3-03719-205-4 print SN - 978-3-03719-705-9 online U6 - https://doi.org/10.4171/205 SP - 85 EP - 132 PB - European Mathematical Society Publishing House CY - Zürich ER - TY - GEN A1 - Paycha, Sylvie T1 - Interview with Pierre Cartier BT - Potsdam, December 4th, 2014 T2 - The mathematical intelligencer Y1 - 2016 U6 - https://doi.org/10.1007/s00283-016-9673-y SN - 0343-6993 SN - 1866-7414 VL - 39 SP - 15 EP - 21 PB - Springer CY - New York ER - TY - JOUR A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - Algebraic Birkhoff factorization and the Euler–Maclaurin formula on cones JF - Duke mathematical journal N2 - We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties-reminiscent of the inclusion-exclusion principle for the cardinal on finite sets-of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the exponential generating sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. A direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an Euler-Maclaurin formula. The compatibility with subdivisions of the convolution quotient arising in the algebraic Birkhoff factorization then yields the Euler-Maclaurin formula for any lattice cone. This provides a simple formula for the interpolating factor by means of a projection formula. Y1 - 2017 U6 - https://doi.org/10.1215/00127094-3715303 SN - 0012-7094 SN - 1547-7398 VL - 166 IS - 3 SP - 537 EP - 571 PB - Duke Univ. Press CY - Durham ER - TY - JOUR A1 - Paycha, Sylvie T1 - When the market wins over research and higher education JF - Sustainable Futures for Higher Education : the Making of Knowledge Makers N2 - In this chapter, an overview of systematic eradication of basic science foci in European universities in the last two decades is given. This happens under the slogan of optimisation of the university education to the needs and demands of the society. It is pointed out that reliance on “market demands” brings with it long-term deficiencies in the maintenance of basic and advanced knowledge construction in societies necessary for long-term future technological advances. University policies that claim improvement of higher education towards more immediate efficiency may end up with the opposite effect of affecting its quality and long term expected positive impact on society. Y1 - 2018 SN - 978-3-319-96035-7 SN - 978-3-319-96034-0 U6 - https://doi.org/10.1007/978-3-319-96035-7_2 SN - 2364-6799 VL - 7 SP - 23 EP - 28 PB - Springer CY - Cham ER - TY - JOUR A1 - Azzali, Sara A1 - Paycha, Sylvie T1 - Spectral zeta-invariants lifted to coverings JF - Transactions of the American Mathematical Society N2 - The canonical trace and the Wodzicki residue on classical pseudo-differential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral zeta-invariants using lifted defect formulae which express discrepancies of zeta-regularised traces in terms of Wodzicki residues. We derive Atiyah's L-2-index theorem as an instance of the Z(2)-graded generalisation of the canonical lift of spectral zeta-invariants and we show that certain lifted spectral zeta-invariants for geometric operators are integrals of Pontryagin and Chern forms. Y1 - 2020 U6 - https://doi.org/10.1090/tran/8067 SN - 0002-9947 SN - 1088-6850 VL - 373 IS - 9 SP - 6185 EP - 6226 PB - American Mathematical Society CY - Providence, RI ER - TY - JOUR A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - Conical zeta values and their double subdivision relations JF - Advances in mathematics N2 - We introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values. In order to achieve the closed cone subdivision relation, we also interpret linear relations among fractions as subdivisions of decorated closed cones. As a generalization of the double shuffle relation of multiple zeta values, we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values. KW - Convex cones KW - Conical zeta values KW - Smooth cones KW - Decorated cones KW - Subdivisions KW - Multiple zeta values KW - Shuffles KW - Quasi-shuffles KW - Fractions with linear poles KW - Shintani zeta values Y1 - 2014 U6 - https://doi.org/10.1016/j.aim.2013.10.022 SN - 0001-8708 SN - 1090-2082 VL - 252 SP - 343 EP - 381 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Levy, Cyril A1 - Jimenez, Carolina Neira A1 - Paycha, Sylvie T1 - THE CANONICAL TRACE AND THE NONCOMMUTATIVE RESIDUE ON THE NONCOMMUTATIVE TORUS JF - Transactions of the American Mathematical Society N2 - Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp. any) noninteger order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace-class operators generalises Fathizadeh and Wong's characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the $ \zeta $-function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence. Y1 - 2016 U6 - https://doi.org/10.1090/tran/6369 SN - 0002-9947 SN - 1088-6850 VL - 368 SP - 1051 EP - 1095 PB - American Mathematical Soc. CY - Providence ER - TY - JOUR A1 - Mickelsson, Jouko A1 - Paycha, Sylvie T1 - The logarithmic residue density of a generalized Laplacian JF - Journal of the Australian Mathematical Society N2 - We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula. KW - residue KW - index KW - Dirac operators Y1 - 2011 U6 - https://doi.org/10.1017/S144678871100108X SN - 0263-6115 SN - 1446-8107 VL - 90 IS - 1 SP - 53 EP - 80 PB - Cambridge Univ. Press CY - Cambridge ER - TY - JOUR A1 - Clavier, Pierre A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - Locality and renormalization: universal properties and integrals on trees JF - Journal of mathematical physics N2 - The purpose of this paper is to build an algebraic framework suited to regularize branched structures emanating from rooted forests and which encodes the locality principle. This is achieved by means of the universal properties in the locality framework of properly decorated rooted forests. These universal properties are then applied to derive the multivariate regularization of integrals indexed by rooted forests. We study their renormalization, along the lines of Kreimer's toy model for Feynman integrals. Y1 - 2020 U6 - https://doi.org/10.1063/1.5116381 SN - 0022-2488 SN - 1089-7658 VL - 61 IS - 2 PB - American Institute of Physics CY - College Park, Md. ER - TY - GEN A1 - Mickelsson, Jouko A1 - Paycha, Sylvie T1 - The logarithmic residue density of a generalized Laplacian T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 649 KW - residue KW - index KW - Dirac operators Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-413680 SN - 1866-8372 IS - 649 ER - TY - JOUR A1 - Bellingeri, Carlo A1 - Friz, Peter A1 - Paycha, Sylvie A1 - Preiß, Rosa Lili Dora T1 - Smooth rough paths, their geometry and algebraic renormalization JF - Vietnam journal of mathematics N2 - We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting. KW - Signatures KW - Rough paths KW - Cartan's development KW - Renormalization Y1 - 2022 U6 - https://doi.org/10.1007/s10013-022-00570-7 SN - 2305-221X SN - 2305-2228 VL - 50 IS - 3 SP - 719 EP - 761 PB - Springer CY - Singapore ER -